The Elegant Proof of Why the Determinant of Matrix Products Equals the Product of Their Determinants

Introduction

One of the fundamental properties in linear algebra is that the determinant of the product of two matrices is equal to the product of their individual determinants. This property, denoted as det(AB) det(A) * det(B), plays a crucial role in various mathematical computations and applications.

While there are multiple proofs for this theorem, one of the most elegant and efficient methods involves leveraging the properties of determinants and using elementary matrices. This article delves into the mathematical elegance and efficiency of this proof, making it accessible and easy to understand.

Proof Using Elementary Matrices

To prove that det(AB) det(A) * det(B), we start by considering simpler cases and then extend the proof to the general case. Specifically, we first prove the theorem when one of the matrices is an elementary matrix and then extend it to the general case using properties of determinants.

Step 1: Proving for Elementary Matrices

An elementary matrix is a matrix that can be obtained by performing a single elementary row or column operation on the identity matrix. The following types of elementary row operations can be applied:

Multiplying a row by a a scalar multiple of one row to another.Switching two rows.

Each of these operations has a well-defined effect on the determinant of a matrix. Let's briefly review the properties:

Multiplying a row by a scalar λ results in a determinant multiplied by λ.Addition of a scalar multiple of one row to another does not change the determinant.Switching two rows changes the sign of the determinant.

Now, consider the determinant of an elementary matrix E. Since E can be derived from the identity matrix by a single row operation, we can apply the above properties to determine the determinant of E. This step is straightforward and relies on the fundamental properties of determinants and row operations.

Step 2: General Case Using Properties of Determinants

Next, we extend the proof to the general case where neither A nor B is necessarily an elementary matrix. We utilize the fact that any square matrix can be expressed as a product of elementary matrices. Therefore, we can represent A and B as follows:

A E1E2...Em, where each Ei is an elementary matrix.

B F1F2...Fn, where each Fj is an elementary matrix.

Using the property that the determinant of a product of matrices is the product of their determinants, we can write:

det(AB) det(E1E2...EmF1F2...Fn) det(E1) * det(E2) * ... * det(Em) * det(F1) * det(F2) * ... * det(Fn) det(A) * det(B).

This brings us to the heart of the proof, showing that the determinant of the product of two matrices is indeed the product of their individual determinants.

Verification via a 2 x 2 Matrix Example

To further illustrate the proof, let's consider a specific example with 2 x 2 matrices A and B:

Matrix A [begin{bmatrix} e f g h end{bmatrix}]

Matrix B [begin{bmatrix} w x y z end{bmatrix}]

The determinants of A and B are calculated as:

det(A) eh - fg

det(B) wz - xy

Next, we find the product AB:

AB [begin{bmatrix} ew fy ex fz gw hy gx hz end{bmatrix}]

The determinant of AB is:

det(AB) (ew fy)(gx hz) - (ex fz)(gw hy)

Expanding this expression, we get:

det(AB) ewgx ewhz fygx fyhz - exgw - exhy - fzgw - fzhy

Simplifying further, we obtain:

det(AB) ewhz fygx - exhy - fzgw (eh - fg)(wz - xy) det(A) * det(B)

This specific example verifies the general property that the determinant of the product of two matrices is the product of their determinants. The steps involved in this computation are systematic and align with the properties of determinants and the distributive law.

Conclusion

In conclusion, the property det(AB) det(A) * det(B) is a fundamental result in linear algebra that can be proven using the properties of determinants and elementary matrices. By understanding these foundations, one can gain a deeper appreciation for the elegance and efficiency of this proof. This property is widely applicable in various mathematical and real-world scenarios, making it a cornerstone of advanced mathematical studies.